Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-field diffusion

Quenching of oscillations via attenuated coupling for dissimilar electrochemical systems

The coupled dynamics of two similar and disparate electrochemical cells oscillators are analyzed. For the similar case, the cells are intentionally operated at different system parameters such that they exhibit distinct oscillatory dynamics ranging from periodic to chaotic. It is observed that when such systems are subjected to an attenuated coupling, implemented bidirectionally, they undergo a mutual quenching of oscillations. The same holds true for the configuration wherein two entirely different electrochemical cells are coupled via bidirectional attenuated coupling. Therefore, the attenuated coupling protocol seems to be universally efficient in achieving oscillation suppression in coupled oscillators (similar or heterogeneous oscillators). The experimental observations were verified by numerical simulations using appropriate electrodissolution model systems. Our results indicate that quenching of oscillations via attenuated coupling is robust and therefore could be ubiquitous in coupled systems with a large spatial separation prone to transmission losses.

Additional complex conjugate feedback-induced explosive death and multistabilities

Many natural and man-made systems require suitable feedback to function properly. In this study, we aim to investigate the impact of additional complex conjugate feedback on globally coupled Stuart-Landau oscillators. We find that this additional feedback results in the onset of symmetry breaking clusters and out-of-phase clusters. Interestingly, we also find the existence of explosive amplitude death along with disparate multistable states. We characterize the first-order transition to explosive death through the amplitude order parameter and show that the transition from oscillatory to death state indeed shows a hysteresis nature. Further, we map the global dynamical transitions in the parametric spaces. In addition, to understand the existence of multistabilities and their transitions, we analyze the bifurcation scenarios of the reduced model and also explore their basin stability. Our study will shed light on the emergent dynamics in the presence of additional feedback.

Effects of propagation delay in coupled oscillators under direct-indirect coupling: Theory and experiment

Propagation delay arises in a coupling channel due to the finite propagation speed of signals and the dispersive nature of the channel. In this paper, we study the effects of propagation delay that appears in the indirect coupling path of direct (diffusive)-indirect (environmental) coupled oscillators. In sharp contrast to the direct coupled oscillators where propagation delay induces amplitude death, we show that in the case of direct-indirect coupling, even a small propagation delay is conducive to an oscillatory behavior. It is well known that simultaneous application of direct and indirect coupling is the general mechanism for amplitude death. However, here we show that the presence of propagation delay hinders the death state and helps the revival of oscillation. We demonstrate our results by considering chaotic time-delayed oscillators and FitzHugh-Nagumo oscillators. We use linear stability analysis to derive the explicit conditions for the onset of oscillation from the death state. We also verify the robustness of our results in an electronic hardware level experiment. Our study reveals that the effect of time delay on the dynamics of coupled oscillators is coupling function dependent and, therefore, highly non-trivial.

Anormal diffusion enhancement of resonant responses for coupled oscillator networks to weak signals

The normal diffusion effect is introduced as a new regulating factor into the established diffusive coupling model for bistable oscillator networks. We find that the response of the system to the weak signal is substantially enhanced by the anormal diffusion, which is termed anormal-diffusion-induced resonance. We also reveal that the diffusive coupling-induced transition, which changes the system from a bistable to a monostable state, is of fundamental importance for the occurrence of resonance. The proposed approach is validated using simulation studies and theoretical analyses. Our results suggest that diffusion induced resonance can be more easily observed in nonlinear oscillator networks.

Oscillating synchronization in delayed oscillators with time-varying time delay coupling: Experimental observation

The time-varying time-delayed (TVTD) systems attract the attention of research communities due to their rich complex dynamics and wide application potentiality. Particularly, coupled TVTD systems show several intriguing behaviors that cannot be observed in systems with a constant delay or no delay. In this context, a new synchronization scenario, namely, oscillating synchronization, was reported by Senthilkumar and Lakshmanan [Chaos 17, 013112 (2007)], which is exclusive to the time-varying time delay systems only. However, like most of the dynamical behavior of TVTD systems, its existence has not been established in an experiment. In this paper, we report the first experimental observation of oscillating synchronization in coupled nonlinear time-delayed oscillators induced by a time-varying time delay in the coupling path. We implement a simple yet effective electronic circuit to realize the time-varying time delay in an experiment. We show that depending upon the instantaneous variation of the time delay, the system shows a synchronization scenario oscillating among lag, complete, and anticipatory synchronization. This study may open up the feasibility of applying oscillating synchronization in engineering systems.

Coexistence of oscillation and quenching states: Effect of low-pass active filtering in coupled oscillators

Effects of a low-pass active filter (LPAF) on the transition processes from oscillation quenching to asymmetrical oscillation are explored for diffusively coupled oscillators. The low-pass filter part and the active part of LPAF exhibit different effects on the dynamics of these coupled oscillators. With the amplifying active part only, LPAF keeps the coupled oscillators staying in a nontrivial amplitude death (NTAD) and oscillation state. However, the additional filter is beneficial to induce a transition from a symmetrical oscillation death to an asymmetrical oscillation death and then to an asymmetrical oscillation state which is oscillating with different amplitudes for two oscillators. Asymmetrical oscillation state is coexisting with a synchronous oscillation state for properly presented parameters. With the attenuating active part only, LPAF keeps the coupled oscillators in rich oscillation quenching states such as amplitude death (AD), symmetrical oscillation death (OD), and NTAD. The additional filter tends to enlarge the AD domains but to shrink the symmetrical OD domains by increasing the areas of the coexistence of the oscillation state and the symmetrical OD state. The stronger filter effects enlarge the basin of the symmetrical OD state which is coexisting with the synchronous oscillation state. Moreover, the effects of the filter are general in globally coupled oscillators. Our results are important for understanding and controlling the multistability of coupled systems.

Transition from homogeneous to inhomogeneous limit cycles: Effect of local filtering in coupled oscillators

We report an interesting symmetry-breaking transition in coupled identical oscillators, namely, the continuous transition from homogeneous to inhomogeneous limit cycle oscillations. The observed transition is the oscillatory analog of the Turing-type symmetry-breaking transition from amplitude death (i.e., stable homogeneous steady state) to oscillation death (i.e., stable inhomogeneous steady state). This novel transition occurs in the parametric zone of occurrence of rhythmogenesis and oscillation death as a consequence of the presence of local filtering in the coupling path. We consider paradigmatic oscillators, such as Stuart-Landau and van der Pol oscillators, under mean-field coupling with low-pass or all-pass filtered self-feedback and through a rigorous bifurcation analysis we explore the genesis of this transition. Further, we experimentally demonstrate the observed transition, which establishes its robustness in the presence of parameter fluctuations and noise.

Quenching oscillating behaviors in fractional coupled Stuart-Landau oscillators

Oscillation quenching has been widely studied during the past several decades in fields ranging from natural sciences to engineering, but investigations have so far been restricted to oscillators with an integer-order derivative. Here, we report the first study of amplitude death (AD) in fractional coupled Stuart-Landau oscillators with partial and/or complete conjugate couplings to explore oscillation quenching patterns and dynamics. It has been found that the fractional-order derivative impacts the AD state crucially. The area of the AD state increases along with the decrease of the fractional-order derivative. Furthermore, by introducing and adjusting a limiting feedback factor in coupling links, the AD state can be well tamed in fractional coupled oscillators. Hence, it provides one an effective approach to analyze and control the oscillating behaviors in fractional coupled oscillators.

Control of bifurcation-delay of slow passage effect by delayed self-feedback

The slow passage effect in a dynamical system generally induces a delay in bifurcation that imposes an uncertainty in the prediction of the dynamical behaviors around the bifurcation point. In this paper, we investigate the influence of linear time-delayed self-feedback on the slow passage through the delayed Hopf and pitchfork bifurcations in a parametrically driven nonlinear oscillator. We perform linear stability analysis to derive the Hopf bifurcation point and its stability as a function of self-feedback time delay. Interestingly, the bifurcation-delay associated with Hopf bifurcation behaves differently in two different edges. In the leading edge of the modulating signal, it decreases with increasing self-feedback delay, whereas in the trailing edge, it behaves in an opposite manner. We also show that the linear time-delayed self-feedback can reduce bifurcation-delay in pitchfork bifurcation. These results are illustrated numerically and corroborated experimentally. We also propose a mechanistic explanation of the observed behaviors. In addition, we show that our observations are robust in the presence of noise. We believe that this study of interplay of two time delays of different origins will shed light on the control of bifurcation-delay and improve our knowledge of time-delayed systems.

Generation and cessation of oscillations: Interplay of excitability and dispersal in a class of ecosystems

We investigate the complex spatiotemporal dynamics of an ecological network with species dispersal mediated via a mean-field coupling. The local dynamics of the network are governed by the Truscott-Brindley model, which is an important ecological model showing excitability. Our results focus on the interplay of excitability and dispersal by always considering that the individual nodes are in their (excitable) steady states. In contrast to the previous studies, we not only observe the dispersal induced generation of oscillation but also report two distinct mechanisms of cessation of oscillations, namely, amplitude and oscillation death. We show that the dispersal between the nodes influences the intrinsic dynamics of the system resulting in multiple oscillatory dynamics such as period-1 and period-2 limit cycles. We also show the existence of multi-cluster states, which has much relevance and importance in ecology.

Oscillation quenching in third order phase locked loop coupled by mean field diffusive coupling

We explored analytically the oscillation quenching phenomena (amplitude death and parameter dependent inhomogeneous steady state) in a coupled third order phase locked loop (PLL) both in periodic and chaotic mode. The phase locked loops were coupled through mean field diffusive coupling. The lower and upper limits of the quenched state were identified in the parameter space of the coupled PLL using the Routh-Hurwitz technique. We further observed that the ability of convergence to the quenched state of coupled PLLs depends on the design parameters. For identical systems, both the systems converge to the homogeneous steady state, whereas for non-identical parameter values they converge to an inhomogeneous steady state. It was also observed that for identical systems, the quenched state is wider than the non-identical case. When the system parameters are so chosen that each isolated loop is chaotic in nature, we observe narrowing down of the quenched state. All these phenomena were also demonstrated through numerical simulations.

Nonlinear behavior of the tarka flute's distinctive sounds

The Andean tarka flute generates multiphonic sounds. Using spectral techniques, we verify two distinctive musical behaviors and the nonlinear nature of the tarka. Through nonlinear time series analysis, we determine chaotic and hyperchaotic behavior. Experimentally, we observe that by increasing the blow pressure on different fingerings, peculiar changes from linear to nonlinear patterns are produced, leading ultimately to quenching.

A common lag scenario in quenching of oscillation in coupled oscillators

A large parameter mismatch can induce amplitude death in two instantaneously coupled oscillators. Alternatively, a time delay in the coupling can induce amplitude death in two identical oscillators. We unify the mechanism of quenching of oscillation in coupled oscillators, either by a large parameter mismatch or a delay coupling, by a common lag scenario that is, surprisingly, different from the conventional lag synchronization. We present numerical as well as experimental evidence of this unknown kind of lag scenario when the lag increases with coupling and at a critically large value at a critical coupling strength, amplitude death emerges in two largely mismatched oscillators. This is analogous to amplitude death in identical systems with increasingly large coupling delay. In support, we use examples of the Chua oscillator and the Bonhoeffer-van der Pol system. Furthermore, we confirm this lag scenario during the onset of amplitude death in identical Stuart-Landau system under various instantaneous coupling forms, repulsive, conjugate, and a type of nonlinear coupling.

Mixed-mode oscillation suppression states in coupled oscillators

We report a collective dynamical state, namely the mixed-mode oscillation suppression state where the steady states of the state variables of a system of coupled oscillators show heterogeneous behaviors. We identify two variants of it: The first one is a mixed-mode death (MMD) state, which is an interesting oscillation death state, where a set of variables show dissimilar values, while the rest arrive at a common value. In the second mixed death state, bistable and monostable nontrivial homogeneous steady states appear simultaneously to a different set of variables (we refer to it as the MNAD state). We find these states in the paradigmatic chaotic Lorenz system and Lorenz-like system under generic coupling schemes. We identify that while the reflection symmetry breaking is responsible for the MNAD state, the breaking of both the reflection and translational symmetries result in the MMD state. Using a rigorous bifurcation analysis we establish the occurrence of the MMD and MNAD states, and map their transition routes in parameter space. Moreover, we report experimental observation of the MMD and MNAD states that supports our theoretical results. We believe that this study will broaden our understanding of oscillation suppression states; subsequently, it may have applications in many real physical systems, such as laser and geomagnetic systems, whose mathematical models mimic the Lorenz system.

Revival of oscillation from mean-field-induced death: Theory and experiment

The revival of oscillation and maintaining rhythmicity in a network of coupled oscillators offer an open challenge to researchers as the cessation of oscillation often leads to a fatal system degradation and an irrecoverable malfunctioning in many physical, biological, and physiological systems. Recently a general technique of restoration of rhythmicity in diffusively coupled networks of nonlinear oscillators has been proposed in Zou et al. [Nat. Commun. 6, 7709 (2015)], where it is shown that a proper feedback parameter that controls the rate of diffusion can effectively revive oscillation from an oscillation suppressed state. In this paper we show that the mean-field diffusive coupling, which can suppress oscillation even in a network of identical oscillators, can be modified in order to revoke the cessation of oscillation induced by it. Using a rigorous bifurcation analysis we show that, unlike other diffusive coupling schemes, here one has two control parameters, namely the density of the mean-field and the feedback parameter that can be controlled to revive oscillation from a death state. We demonstrate that an appropriate choice of density of the mean field is capable of inducing rhythmicity even in the presence of complete diffusion, which is a unique feature of this mean-field coupling that is not available in other coupling schemes. Finally, we report the experimental observation of revival of oscillation from the mean-field-induced oscillation suppression state that supports our theoretical results.

Dispersal-induced synchrony, temporal stability, and clustering in a mean-field coupled Rosenzweig-MacArthur model

In spatial ecology, dispersal among a set of spatially separated habitats, named as metapopulation, preserves the diversity and persistence by interconnecting the local populations. Understanding the effects of several variants of dispersion in metapopulation dynamics and to identify the factors which promote population synchrony and population stability are important in ecology. In this paper, we consider the mean-field dispersion among the habitats in a network and study the collective dynamics of the spatially extended system. Using the Rosenzweig-MacArthur model for individual patches, we show that the population synchrony and temporal stability, which are believed to be of conflicting outcomes of dispersion, can be simultaneously achieved by oscillation quenching mechanisms. Particularly, we explore the more natural coupling configuration where the rates of dispersal of different habitats are disparate. We show that asymmetry in dispersal rate plays a crucial role in determining inhomogeneity in an otherwise homogeneous metapopulation. We further identify an unusual emergent state in the network, namely, a multi-branch clustered inhomogeneous steady state, which arises due to the intrinsic parameter mismatch among the patches. We believe that the present study will shed light on the cooperative behavior of spatially structured ecosystems.

Mean-field dispersion-induced spatial synchrony, oscillation and amplitude death, and temporal stability in an ecological model

One of the most important issues in spatial ecology is to understand how spatial synchrony and dispersal-induced stability interact. In the existing studies it is shown that dispersion among identical patches results in spatial synchrony; on the other hand, the combination of spatial heterogeneity and dispersion is necessary for dispersal-induced stability (or temporal stability). Population synchrony and temporal stability are thus often thought of as conflicting outcomes of dispersion. In contrast to the general belief, in this present study we show that mean-field dispersion is conducive to both spatial synchrony and dispersal-induced stability even in identical patches. This simultaneous occurrence of rather conflicting phenomena is governed by the suppression of oscillation states, namely amplitude death (AD) and oscillation death (OD). These states emerge through spatial synchrony of the oscillating patches in the strong-coupling strength. We present an interpretation of the mean-field diffusive coupling in the context of ecology and identify that, with increasing mean-field density, an open ecosystem transforms into a closed ecosystem. We report on the occurrence of OD in an ecological model and explain its significance. Using a detailed bifurcation analysis we show that, depending on the mortality rate and carrying capacity, the system shows either AD or both AD and OD. We also show that the results remain qualitatively the same for a network of oscillators. We identify a new transition scenario between the same type of oscillation suppression states whose geneses differ. In the parameter-mismatched case, we further report on the direct transition from OD to AD through a transcritical bifurcation. We believe that this study will lead to a proper interpretation of AD and OD in ecology, which may be important for the conservation and management of several communities in ecosystems.

Transitions among the diverse oscillation quenching states induced by the interplay of direct and indirect coupling

We report the transitions among different oscillation quenching states induced by the interplay of diffusive (direct) coupling and environmental (indirect) coupling in coupled identical oscillators. This coupling scheme was introduced by Resmi et al. [Phys. Rev. E 84, 046212 (2011)] as a general scheme to induce amplitude death (AD) in nonlinear oscillators. Using a detailed bifurcation analysis we show that, in addition to AD, which actually occurs only in a small region of parameter space, this coupling scheme can induce other oscillation quenching states, namely oscillation death (OD) and a novel nontrvial AD (NAD) state, which is a nonzero bistable homogeneous steady state; more importantly, this coupling scheme mediates a transition from the AD state to the OD state and a new transition from the AD state to the NAD state. We identify diverse routes to the NAD state and map all the transition scenarios in the parameter space for periodic oscillators. Finally, we present the first experimental evidence of oscillation quenching states and their transitions induced by the interplay of direct and indirect coupling.

Experimental observation of a transition from amplitude to oscillation death in coupled oscillators

We report the experimental evidence of an important transition scenario, namely the transition from amplitude death (AD) to oscillation death (OD) state in coupled limit cycle oscillators. We consider two Van der Pol oscillators coupled through mean-field diffusion and show that this system exhibits a transition from AD to OD, which was earlier shown for Stuart-Landau oscillators under the same coupling scheme [T. Banerjee and D. Ghosh, Phys. Rev. E 89, 052912 (2014)]. We show that the AD-OD transition is governed by the density of mean-field and beyond a critical value this transition is destroyed; further, we show the existence of a nontrivial AD state that coexists with OD. Next, we implement the system in an electronic circuit and experimentally confirm the transition from AD to OD state. We further characterize the experimental parameter zone where this transition occurs. The present study may stimulate the search for the practical systems where this important transition scenario can be observed experimentally.

Transition from amplitude to oscillation death under mean-field diffusive coupling

We study the transition from the amplitude death (AD) to the oscillation death (OD) state in limit-cycle oscillators coupled through mean-field diffusion. We show that this coupling scheme can induce an important transition from AD to OD even in identical limit cycle oscillators. We identify a parameter region where OD and a nontrivial AD (NTAD) state coexist. This NTAD state is unique in comparison with AD owing to the fact that it is created by a subcritical pitchfork bifurcation and parameter mismatch does not support this state, but destroys it. We extend our study to a network of mean-field coupled oscillators to show that the transition scenario is preserved and the oscillators form a two-cluster state.